On The Nonlinear Generalized Langevin Equation Involving ψ-Caputo Fractional Derivatives

This paper considers the generalized Langevin equation involving ψ-Caputo fractional derivatives in a Banach space. The fractional derivative is generalized from the Caputo derivative (ψ(t) = t), the Caputo–Katugampola (ψ(t) = ÏˆÏ (t) = (tÏ âˆ’ 1)/Ï , the Hadamard derivative (ψ(t) = ψH(t) =ln t). We investigate the existence of mild solutions uψ of the problem, in which the source function is assumed to satisfy some weakly singular conditions. Before proceeding to the main results, we transform the problem into an integral equation. Based on the obtained integral equation, the main results are proved via the nonlinear Leray–Schauder alternatives and Banach fixed point theorems. To prove this end, a new generalized weakly Gronwall-type inequality is established. Further, we prove that the mild solution of the problem is dependent continuously on the inputs: initial data, fractional orders and the friction constant. As a consequence, we deduce that the solution uÏˆÏ of the equation involving the Caputo–Katugampola derivative tends to the solution uψH of the equation involving the Hadamard derivative as Ï â†’ 0+.